Hierarchical Comprehensive Triangular Decomposition

The concept of

comprehensive triangular decomposition

(CTD) was first introduced by Chen

et al.

in their CASC’2007 paper and could be viewed as an analogue of comprehensive Gröbner systems for parametric polynomial systems. The first complete algorithm for computing CTD was also proposed in that paper and implemented in the

RegularChains

library in Maple. Following our previous work on generic regular decomposition for parametric polynomial systems, we introduce in this paper a so-called

hierarchical

strategy for computing CTDs. Roughly speaking, for a given parametric system, the parametric space is divided into several sub-spaces of different dimensions and we compute CTDs over those sub-spaces one by one. So, it is possible that, for some benchmarks, it is difficult to compute CTDs in reasonable time while this strategy can obtain some “partial” solutions over some parametric sub-spaces. The program based on this strategy has been tested on a number of benchmarks from the literature. Experimental results on these benchmarks with comparison to

RegularChains

are reported and may be valuable for developing more efficient triangularization tools.